Test of Mathematics for University Admission (TMUA) Equations and Inequalities Syllabus

Equations and Inequalities syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Equations and Inequalities in Test of Mathematics for University Admission (TMUA), not a summary of it.

1. 1

   Linear and Simultaneous Equations

   3 topics

   • Solving Linear Equations
     • Rearranging and changing the subject
     • Equations with fractions
   • Simultaneous Equations
     • Linear-linear systems
     • Linear-quadratic systems
   • Geometric Interpretation
     • Intersection of lines and curves
     • Number of solutions
2. 2

   Quadratic Equations and the Discriminant

   3 topics

   • Solving Quadratics
     • Factorisation method
     • Quadratic formula
     • Completing the square
   • The Discriminant
     • Nature of roots
     • Conditions for real or repeated roots
   • Equations Reducible to Quadratics
     • Disguised quadratics in u-substitution
     • Quadratics in surds and exponentials
3. 3

   Inequalities

   4 topics

   • Linear Inequalities
     • Single and combined inequalities
     • Inequalities with negative multipliers
   • Quadratic Inequalities
     • Sign diagrams and sketch method
     • Critical values and intervals
   • Inequalities with Algebraic Fractions
     • Sign analysis of rational expressions
   • The Modulus Function
     • Solving modulus equations
     • Solving modulus inequalities

Equations and Inequalities flashcards for Test of Mathematics for University Admission (TMUA)

22 of 50 cards from the Equations and Inequalities deck — real questions with worked answers.

1. What is the general process for solving a linear equation in one variable?

   Collect all variable terms on one side and constants on the other (using inverse operations applied equally to both sides), simplify to the form $ax = b$, then divide by the coefficient to get $x = \frac{b}{a}$ (provided $a \neq 0$).

2. When does the linear equation $ax = b$ have (i) a unique solution, (ii) no solution, (iii) infinitely many solutions?

   (i) Unique solution $x=\frac{b}{a}$ when $a \neq 0$; (ii) no solution when $a = 0$ and $b \neq 0$; (iii) infinitely many solutions when $a = 0$ and $b = 0$.

3. How do you solve a linear equation containing fractions, e.g. $\frac{x}{3} + \frac{x}{4} = 7$?

   Multiply every term by the lowest common multiple of the denominators (here $12$) to clear fractions: $4x + 3x = 84$, giving $7x = 84$, so $x = 12$.

4. What does it mean to solve a pair of simultaneous equations?

   To find the value(s) of the unknowns that satisfy all the equations at the same time; geometrically, the coordinates of the point(s) where the graphs intersect.

5. Describe the elimination method for solving two linear simultaneous equations.

   Scale one or both equations so that the coefficients of one variable are equal in magnitude, then add or subtract the equations to eliminate that variable, solve for the remaining variable, and back-substitute to find the other.

6. Describe the substitution method for solving simultaneous equations.

   Rearrange one equation to make one variable the subject, substitute that expression into the other equation to get a single-variable equation, solve it, then back-substitute to find the remaining variable.

7. Which method is generally preferred for a linear–quadratic pair of simultaneous equations, and why?

   Substitution: rearrange the linear equation for one variable and substitute into the quadratic, producing a single quadratic equation in one variable to solve.

8. For two linear equations, what determines whether they have one solution, no solution, or infinitely many solutions?

   One solution: the lines have different gradients (intersect once). No solution: same gradient, different intercepts (parallel, distinct). Infinitely many: same gradient and same intercept (identical lines).

9. Geometrically, what does the solution of a linear–quadratic system represent, and how many solutions are possible?

   The intersection points of a line and a parabola (or other conic). There can be $2$ solutions (line cuts the curve), $1$ solution (line is tangent), or $0$ solutions (line misses the curve).

10. How does the discriminant of the resulting quadratic tell you the geometric relationship between a line and a curve?

    After substituting, the quadratic's discriminant $b^{2}-4ac$ gives: $>0$ two intersection points (secant), $=0$ one point (tangent), $<0$ no intersection.

11. State the quadratic formula for solving $ax^{2}+bx+c=0$.

    $$x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}, \quad a \neq 0.$$

12. What is the method of completing the square for $x^{2}+bx+c$?

    Write $x^{2}+bx+c = \left(x+\frac{b}{2}\right)^{2} - \left(\frac{b}{2}\right)^{2} + c$, halving the coefficient of $x$ and subtracting the square of that half.

13. How do you solve a quadratic equation by factorising?

    Write it as $ax^{2}+bx+c=0$, factorise into $(px+q)(rx+s)=0$, then set each factor to zero (using the null factor law) and solve, giving $x=-\frac{q}{p}$ or $x=-\frac{s}{r}$.

14. For $ax^{2}+bx+c=0$ with roots $\alpha$ and $\beta$, what are the sum and product of the roots?

    $$\alpha+\beta = -\frac{b}{a}, \qquad \alpha\beta = \frac{c}{a}.$$

15. What is the formula for the discriminant of $ax^{2}+bx+c=0$, and what does it determine?

    The discriminant is $\Delta = b^{2}-4ac$. It determines the number and nature of the real roots of the quadratic.

16. Classify the roots of a quadratic according to the sign of the discriminant $b^{2}-4ac$.

    $b^{2}-4ac>0$: two distinct real roots. $b^{2}-4ac=0$: one repeated (equal) real root. $b^{2}-4ac<0$: no real roots (two complex conjugate roots).

17. What condition on the discriminant guarantees a quadratic has two distinct real roots?

    $b^{2}-4ac > 0$.

18. What does $b^{2}-4ac=0$ tell you about a parabola $y=ax^{2}+bx+c$ and the $x$-axis?

    The parabola touches the $x$-axis at exactly one point; the $x$-axis is tangent to it (a repeated root).

19. If a quadratic $ax^{2}+bx+c$ is a perfect square, what is the value of its discriminant?

    $b^{2}-4ac = 0$, since a perfect square has a repeated root.

20. For what values of $k$ does $x^{2}+kx+9=0$ have equal roots?

    Equal roots require $\Delta=0$: $k^{2}-4(1)(9)=0$, so $k^{2}=36$, giving $k=\pm 6$.

21. What is meant by an equation 'reducible to a quadratic', and how do you solve one?

    An equation that becomes quadratic after a suitable substitution. Substitute $u$ for a repeated expression to get a quadratic in $u$, solve for $u$, then reverse the substitution to find the original variable.

22. How do you solve a quartic of the form $ax^{4}+bx^{2}+c=0$?

    Let $u=x^{2}$ to get $au^{2}+bu+c=0$, solve for $u$, then solve $x^{2}=u$ for each root, taking $x=\pm\sqrt{u}$ (real only where $u \geq 0$).

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